Conditions for the Particle to Fall on the Center of the Potential

To determine the conditions under which the particle will fall on the center of the potential, we need to analyze the behavior of the potential energy function and the angular momentum of the particle.

1. Potential Energy Function

The potential energy function for the given potential V(r) = a/r^2 represents an inverse square law potential. This type of potential is characteristic of central forces, such as the gravitational or electrostatic forces.

When a particle is subjected to an inverse square law potential, its trajectory will depend on the total mechanical energy of the system. In this case, the mechanical energy E is given by the sum of the kinetic energy and the potential energy:

E = 1/2 * μ * v^2 + a/r^2

Where μ is the mass of the particle and v is its velocity.

2. Angular Momentum

The angular momentum of the particle is given by L = μ * r * v, where r is the distance of the particle from the center of the potential.

3. Conditions for Falling on the Center

For the particle to fall on the center of the potential, two conditions must be satisfied:

- The total mechanical energy E must be negative, indicating a bound state where the particle is confined to a finite region. This ensures that the particle will not escape to infinity.

- The angular momentum L must be zero. This condition implies that there is no rotational motion around the center of the potential, leading the particle to move directly towards the center.

Conclusion for the Coulomb Problem

The Coulomb problem involves a point charge interacting with another point charge through the electrostatic force, which is an inverse square law potential. By applying the conditions discussed above, we can draw the following conclusions:

- In the Coulomb problem, the particle will fall on the center of the potential if its total mechanical energy is negative and its angular momentum is zero.

- The negative total mechanical energy ensures that the particle is bound to the center and will not escape to infinity.

- The zero angular momentum implies that there is no rotational motion around the center, leading the particle to move directly towards the center.

- This behavior is consistent with the physical intuition of the Coulomb problem, where a charged particle experiences an attractive force towards the center due to the opposite charges.

In conclusion, the conditions for a particle to fall on the center of the potential involve a negative total mechanical energy and zero angular momentum. These conditions hold for the Coulomb problem, where a charged particle falls towards the center due to the attractive electrostatic force.